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Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra ACM Communications in Computer Algebra, Vol. 50, No. 1, Issue 195, March 2016 Workshop on Symbolic Combinatorics and Algorithmic Di↵erential Algebra Manuel Kauers1,PeterPaule2,GregReid3 1 Instiute for Algebra, Johannes Kepler University, Austria 2 Research Institute for Symbolic Computation, Johannes Kepler University, Austria 3 University of Western Ontario, Canada The workshop on Symbolic Combinatorics and Computational Di↵erential Algebra was held from Septem- ber 14 to September 20, 2015 as part of the Thematic Program on Computer Algebra at the Fields Institute in Toronto, Canada. The workshop was devoted to algorithmic developments in Combinatorics and Di↵erential Algebra with a particular focus on the interaction of these two areas. Symbolic Combinatorics: Symbolic algorithms and software have recently been developed that allow researchers to discover and prove combinatorial identities as well as understand analytic and algebraic properties of generating functions. These functions seldom have closed form solutions and even when they do, direct evaluation may be intractable. Recent work in Symbolic Combinatorics has focused on representing such functions by annihilating dif- ferential or di↵erence operators and then using techniques from di↵erential and di↵erence algebra, as well as analysis, to analyze these equations and their solutions. Besides methods relating to creative telescoping and verication of function identities, other areas of emphasis will include e↵ective methods in di↵erence algebra, the Galois theories of di↵erence equations, and the interaction of di↵erential and di↵erence algebra in combinatorics. Computational Di↵erential Algebra is an approach to nonlinear or linear di↵erential equations and di↵erential struc- tures, focusing not only on finding explicit closed form solution, but also on simplifying such equations to yield preconditioning for subsequent numerical solution, as well clearer understanding of the qualitative behavior of solu- tions. Highlighted aspects will include di↵erential-elimination completion algorithms; geometric di↵erential algebra, moving frames and di↵erential invariants; di↵erential Galois theory and integrability; complexity and open problems. We have greatly enjoyed the diverse and inspiring contributions from the speakers who contributed to the program. Their abstracts are quoted below. Carlos Arreche, North Carolina State University On the computation of the di↵erence-di↵erential Galois group for a second-order linear di↵erence equation Abstract. Given a linear di↵erence equation, there is a di↵erence-di↵erential Galois group that encodes the di↵erential- algebraic dependencies among the solutions of the equation. After giving a brief introduction to this theory, I will describe algorithms to compute the Galois group associated to a second-order linear di↵erence equation over C(x), the field of rational functions over a computable field C of characteristic zero, with respect to the C-linear shift automorphism that sends x to x + 1. I will also discuss some concrete examples to illustrate these algorithms, and show explicitly in the examples how to derive the di↵erential-algebraic dependencies among the solutions from the knowledge of the defining equations for the Galois group. Cyril Banderier, Institut Galil´ee– Universit´eParis-Nord From algebraic to di↵erentialy-algebraic functions in combinatorics: impact of positive (integer) co- efficients Abstract. Asymptotics of recurrences is the key to get the typical properties of combinatorial structures, and thus the complexity of many algorithms relying on these structures. The associated generating function often follows a linear di↵erential equation: we are here in the so-called “D-finite” world. For the matters of asymptotics, this case of 27 WSCADA Abstracts linear recurrences (with polynomial coefficients) is well covered by the “Analytic Combinatorics” book of Flajolet and Sedgewick (though the computations of constants is still a challenge, related to the theory of Kontsevich-Zagier pe- riods and evaluation of G-functions and E-functions). At the border of this D-finite world, lies “algebraic-di↵erential functions”. The terminology is not yet fixed and similar terms are used, up to a permutation, by several authors: let dzm be the m-th derivative of F (z), the function is said “algebraic-di↵erential” if there a exists a polynomial P m such that P (z,F,F 0,...,dz F ) = 0. For all these worlds, having some positive (integer) coefficients leads to some strong constraints on the asymptotics (these is now well understood for algebraic function), and we try to see what happens in a more general setting. We will give examples of such functions (motivated by some combinatorial problems), and show how a symbolic combinatorics approach can help for automatic asymptotics of their coefficients, and some open related open ques- tions/challenges for computer algebra (joint works with Michael Drmota and Hsien-Kuei Hwang) Moulay A. Barkatou, Universite de Limoges On Apparent Singularities of Systems of Linear Di↵erential Equations with Rational Function Coef- ficients dY Abstract. Let (S) dz = A(z)Y be a system of first order linear di↵erential equations with rational function coeffi- cients. The (finite) singularities of (S) are the poles of the entries de the matrix A(z). A singular point z0 of (S) is called an apparent singularity for (S) if there is a basis of solutions of (S) which are holomorphic in a neighborhood of z0. In this talk we shall present a new algorithm which, given a system of the form (S), detects apparent singu- larities and constructs an equivalent system (S’) with rational coefficients the singularities of which coincide with the non apparent singularities of (S). Our method can, in particular, be applied to the companion system of any linear di↵erential equation with arbitrary order n. We thus have an alternative method to the standard methods for removing apparent singularities of linear di↵erential operators. We shall compare our method to the one designed for operators and we shall show some applications and examples of computation. This talk is based on a joint work, with S. Maddah, recently presented at ISSAC 2015. Alin Bostan, INRIA Quasi-optimal computation of the p-curvature Abstract. The p-curvature of a system of linear di↵erential equations in characteristic p is a matrix that measures to what extent the system is close to having a fundamental matrix of rational function solutions. This notion, originally introduced in the arithmetic theory of di↵erential equations, has been recently used as an e↵ective tool in computer algebra and in combinatorial applications. We describe a recent algorithm for computing the p-curvature, whose complexity is almost optimal with respect to the size of the output. The new algorithm performs remarkably well in practice. Its design relies on the existence of a well-suited ring, of so-called Hurwitz series, for which an analogue of the Cauchy–Lipschitz Theorem holds, and on a FFT-like method in which the “evaluation points” are Hurwitz series. Joint work with Xavier Caruso and Eric´ Schost. Fr´ed´eric Chyzak, INRIA Saclay Ile-de-Franceˆ Explicit generating series for small-step walks in the quarter plane Abstract. Lattice walks occur frequently in discrete mathematics, statistical physics, probability theory, and opera- tional research. The algebraic properties of their enumeration generating series vary greatly according to the family of admissible steps chosen to define them: their generating series are sometimes rational, algebraic, or D-finite, or sometimes they possess no apparent equation. This has recently motivated a large classification e↵ort. Interestingly, the equations involved often have degrees, orders, and sizes, making calculations an interesting challenge for computer algebra. In this talk, we study nearest-neighbours walks on the square lattice, that is, models of walks on the square lattice, defined by a fixed step set that is a subset of the 8 non-zero vectors with coordinates 0 or 1. We concern ourselves with the counting of walks constrained to remain in the quarter plane, counted by length.± In the past, Bousquet- M´elou and Mishna showed that only 19 essentially di↵erent models of walks possess a non-algebraic D-finite generating series; the linear di↵erential equations have then been guessed by Bostan and Kauers. In this work in progress, we give the first proof that these equations are satisfied by the corresponding generating series. This allows to derive nice formulas for the generating series, expressed in terms of Gauss’ hypergeometric series, to decide their algebraicity or transcendence. This also gives hope to extract asymptotic formulas for the number of walks counted by lengths. 28 M. Kauers, P. Paule & G. Reid (Based on work in progress with Alin Bostan, Mark van Hoeij, Manuel Kauers, and Lucien Pech.) Shaoshi Chen, Chinese Academy of Sciences Proof of the Wilf-Zeilberger Conjecture Abstract. In 1992, Wilf and Zeilberger conjectured that a hypergeometric term in several discrete and continuous variables is holonomic if and only if it is proper. Strictly speaking the conjecture does not hold, but it is true when reformulated properly: Payne proved a piecewise interpretation in 1997, and independently, Abramov and Petkovsek in 2002 proved a conjugate interpretation. Both results address the pure discrete case of the conjecture. In this paper we extend their work to hypergeometric terms in several discrete and continuous variables and prove the conjugate interpretation of the Wilf-Zeilberger conjecture in this mixed setting. This is a joint work with Christoph Koutschan. Evelyne Hubert, INRIA M´editerran´ee;Coauthor: Mathieu Collowald A moment matrix approach to symmetric cubatures Abstract. A quadrature is an approximation of the definite integral of a function by a weighted sum of function values at specified points, or nodes, within the domain of integration. Gaussian quadratures are constructed to yield exact results for any polynomials of degree 2r 1 or less by a suitable choice of r nodes and weights.
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